A human with certain types of supernatural powers, Most commonly one or more of the following:
1. Extra Scensory Perception (esp)
2. Power over the four elements (Fire, Earth, Water, Air)
3. Time Travel
4. Power to manipulate space and matter
Esper activities are controled by the "Agency", I'm not sure of what this agency is made of but they have the power to control nearly every aspect of the material world.
Esper powers are no controled by the espers will, certain conditions must be met (as to what im not sure but i will update eventually).

Human: Alright, I will believe you're an esper if you can heat this cup of coffee.

Esper: My powers aren't that easy to capture, certain conditions apply, but as I mentioned there will be plenty of time for me to show you my powers later...

Final fantasy 3/6(same game) esper's are the magical beings, with surprise, magic, after the war of the magi, they made their own world and hide in it, to avoid giving ou tthe story to anyone, ill stop the definition there